A Computational Verification Method of Existence of Solutions for Nonlinear Elliptic Equations

Mitsuhiro T. Nakao

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

This chapter discusses a computational verification method of existence of solutions for nonlinear elliptic equations. It proposes a numerical method for automatic proof of the existence of weak solutions for certain linear elliptic boundary value problems by computer. And its extension to the more general linear case is described. The main techniques in these works consist of the verification method by computer for the existential condition of solutions based on the infinite dimensional fixed point theorems. This chapter uses the properties of the solution for Poisson's equation and the results of error estimates for the finite element approximation as well as the method of interval arithmetic. This chapter formulates a numerical verification method which can be applicable to nonlinear elliptic boundary value problems.

Original languageEnglish
Pages (from-to)101-120
Number of pages20
JournalNorth-Holland Mathematics Studies
Volume160
Issue numberC
DOIs
Publication statusPublished - 1989 Jan 1
Externally publishedYes

Fingerprint

Nonlinear Elliptic Equations
Existence of Solutions
Nonlinear Elliptic Boundary Value Problem
Numerical Verification
Interval Arithmetic
Existence of Weak Solutions
Elliptic Boundary Value Problems
Finite Element Approximation
Poisson's equation
Fixed point theorem
Error Estimates
Numerical Methods

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

A Computational Verification Method of Existence of Solutions for Nonlinear Elliptic Equations. / Nakao, Mitsuhiro T.

In: North-Holland Mathematics Studies, Vol. 160, No. C, 01.01.1989, p. 101-120.

Research output: Contribution to journalArticle

@article{ac2af9f254594e8d97cca6e485876303,
title = "A Computational Verification Method of Existence of Solutions for Nonlinear Elliptic Equations",
abstract = "This chapter discusses a computational verification method of existence of solutions for nonlinear elliptic equations. It proposes a numerical method for automatic proof of the existence of weak solutions for certain linear elliptic boundary value problems by computer. And its extension to the more general linear case is described. The main techniques in these works consist of the verification method by computer for the existential condition of solutions based on the infinite dimensional fixed point theorems. This chapter uses the properties of the solution for Poisson's equation and the results of error estimates for the finite element approximation as well as the method of interval arithmetic. This chapter formulates a numerical verification method which can be applicable to nonlinear elliptic boundary value problems.",
author = "Nakao, {Mitsuhiro T.}",
year = "1989",
month = "1",
day = "1",
doi = "10.1016/S0304-0208(08)70508-5",
language = "English",
volume = "160",
pages = "101--120",
journal = "North-Holland Mathematics Studies",
issn = "0304-0208",
publisher = "Elsevier",
number = "C",

}

TY - JOUR

T1 - A Computational Verification Method of Existence of Solutions for Nonlinear Elliptic Equations

AU - Nakao, Mitsuhiro T.

PY - 1989/1/1

Y1 - 1989/1/1

N2 - This chapter discusses a computational verification method of existence of solutions for nonlinear elliptic equations. It proposes a numerical method for automatic proof of the existence of weak solutions for certain linear elliptic boundary value problems by computer. And its extension to the more general linear case is described. The main techniques in these works consist of the verification method by computer for the existential condition of solutions based on the infinite dimensional fixed point theorems. This chapter uses the properties of the solution for Poisson's equation and the results of error estimates for the finite element approximation as well as the method of interval arithmetic. This chapter formulates a numerical verification method which can be applicable to nonlinear elliptic boundary value problems.

AB - This chapter discusses a computational verification method of existence of solutions for nonlinear elliptic equations. It proposes a numerical method for automatic proof of the existence of weak solutions for certain linear elliptic boundary value problems by computer. And its extension to the more general linear case is described. The main techniques in these works consist of the verification method by computer for the existential condition of solutions based on the infinite dimensional fixed point theorems. This chapter uses the properties of the solution for Poisson's equation and the results of error estimates for the finite element approximation as well as the method of interval arithmetic. This chapter formulates a numerical verification method which can be applicable to nonlinear elliptic boundary value problems.

UR - http://www.scopus.com/inward/record.url?scp=77956935685&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956935685&partnerID=8YFLogxK

U2 - 10.1016/S0304-0208(08)70508-5

DO - 10.1016/S0304-0208(08)70508-5

M3 - Article

AN - SCOPUS:77956935685

VL - 160

SP - 101

EP - 120

JO - North-Holland Mathematics Studies

JF - North-Holland Mathematics Studies

SN - 0304-0208

IS - C

ER -